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# Toric

## toric varieties and some combinatorial geometry computations

1.9.6

4 July 2024

David Joyner
Email: wdjoyner@gmail.com
Homepage: https://sites.google.com/site/wdjoyner/
Address:
W. David Joyner
Mathematics Department
U. S. Naval Academy
Annapolis, MD 21402
USA

### Copyright

© 2004-2017 David Joyner.

### Acknowledgements

The code for the toric package was written during the summer of 2002. It was put into GAP package format in the summer of 2004.

toric is free software; you can redistribute it and/or modify it under the terms of the MIT License.

toric is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the MIT License for more details.

This documentation was prepared with the GAPDoc package of Frank Lübeck and Max Neunhöffer. Moreover, a bug in toric 1.8 was fixed with the help of Max Horn, and this documentation was modified accordingly. Finally, I thank Olexandr Konovalov and Max Horn for transferring this package to the new Git repository.

### Contents

1 Introduction
1.1 Introduction to the toric package
1.2 Introduction to constructing toric varieties
1.2-1 Generalities
1.2-2 Basic combinatorial geometry constructions
1.2-3 Basic affine toric variety constructions
1.2-4 Riemann-Roch spaces and related constructions
2 Cones and semigroups
2.1 Cones
2.1-1 InsideCone
2.1-2 InDualCone
2.1-3 PolytopeLatticePoints
2.1-4 Faces
2.1-5 ConesOfFan
2.1-6 NumberOfConesOfFan
2.1-7 ToricStar
2.2 Semigroups
2.2-1 DualSemigroupGenerators
3 Affine toric varieties
3.1 Ideals defining affine toric varieties
3.1-1 EmbeddingAffineToricVariety
4 Toric varieties \(X(\Delta)\)
4.1 Riemann-Roch spaces
4.1-1 DivisorPolytope
4.1-2 DivisorPolytopeLatticePoints
4.1-3 RiemannRochBasis
4.2 Topological invariants
4.2-1 EulerCharacteristic
4.2-2 BettiNumberToric
4.3 Points over a finite field
4.3-1 CardinalityOfToricVariety
References
Index
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